Download On some mappings in topological spaces

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 1, No. 4, 2008, (22-29)
ISSN 1307-5543 – www.ejpam.com
On some mappings in topological spaces
Bashir Ahmad1 , Sabir Hussain2,∗ , Takashi Noiri3
1
Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University,
Multan, Pakistan.
Present Address: Department of Mathematics, King Abdul Aziz University P. O. Box 80203, Jeddah
21589, Saudi Arabia.
2
Department of Mathematics, Islamia University Bahawalpur, Pakistan.
3
2949-1 Shiokita-cho, Hinagu, Yatsushiro-shi, Kumamoto-ken, 869-5142, Japan.
Abstract. In this paper, we continue studying the properties of γ-semi-continuous and γ -semi-open
functions introduced in [5], [10]. We also introduce γ∗ -irresolute functions and γ -pre-semi-open
(closed) functions and discuss their properties.
AMS subject classifications: 54A05, 54A10, 54D10.
Key words: γ-closed (open), γ-closure , γ∗ -semi-closed (open), γ∗ -semi-closure, γ-regular, γ∗ -semiinterior, γ-semi-continuous functions, γ-semi-open (closed) functions, γ∗ -irresolute functions, γ-presemi-open(closed) functions.
1. Introduction
A. Csaszar [7,8] defined generalized open sets in generalized topological spaces. In 1975,
Maheshwari and Prasad [13] introduced concepts of semi- T1 -spaces and semi-R0 -spaces. In
1979, S. Kasahara [11] defined an operation α on topological spaces. In 1992 (1993), B.
Ahmad and F.U. Rehman [1], [15] introduced the notions of γ-interior, γ-boundary and γexterior points in topological spaces. They also studied properties and characterizations of
(γ, β)-continuous mappings introduced by H. Ogata [14]. In 1999 (resp. 2005), B. Ahmad
and S. Hussain introduced the concept of γ∗ -regular spaces ( resp. γ0 -compact, γ-normal
spaces) and explored their many interesting properties. They initiated and discussed the
concepts of γ∗ -semi-open sets which generalizes γ-open sets introduced and discussed by H.
Ogata [14], γ∗ -semi-closed sets, γ∗ -semi-closure, γ∗ -semi-interior point in topological spaces
γ
γ
[5], [9]. In 2006, they introduced Λs -set and Λs -set by using γ∗ -semi-open sets. Moreover, they introduced the γ-semi-continuous functions and γ-semi-open (closed) functions in
topological spaces and established several interesting properties.
In this paper, we continue studying the properties of γ-semi-continuous and γ-semi-open
functions introduced by B. Ahmad and S. Hussain [5], [10]. We also introduce γ∗ -irresolute
∗
Corresponding author. Email addresses: [email protected] (B. Ahmad)
[email protected] (S. Hussain), [email protected] (T. Noiri)
http://www.ejpam.com
22
c 2008 EJPAM All rights reserved.
B. Ahmad, S. Hussain , and T. Noiri / Eur. J. Pure Appl. Math, 1 (2008), (22-29)
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functions and γ-pre-semi-open (closed) functions and discuss their properties.
Hereafter, we shall write spaces in place of topological spaces in the sequel.
2. Preliminaries
We recall some definitions and results used in this paper to make it self-contained.
Definition [11]. Let (X,τ) be a space. An operation γ : τ → P(X) is a function from τ to the
power set of X such that V ⊆ V γ , for each V ∈ τ, where V γ denotes the value of γ at V. The
operations defined by γ(G) = G, γ(G) = cl(G) and γ(G) = intcl(G) are examples of operation
γ.
Definition [14]. Let A⊆ X. A point x∈ A is said to be γ-interior point of A iff there exists an
open nbd N of x such that N γ ⊆ A and we denote the set of all such points by int γ (A). Thus
int γ (A) = {x ∈ A : x ∈ N ∈ τ and N γ ⊆ A} ⊆ A.
Note that A is γ-open [14] iff A =int γ (A). A set A is called γ- closed [14] iff X-A is γ-open.
Definition [14]. A point x∈ X is called a γ-closure point of A⊆ X, if U γ ∩ A 6= φ, for each
open nbd U of x. The set of all γ-closure points of A is called γ-closure of A and is denoted by
clγ (A). A subset A of X is called γ-closed, if clγ (A) ⊆A. Note that clγ (A) is contained in every
γ-closed superset of A.
Definition [14]. An operation γ on τ is said be regular, if for any open nbds U,V of x ∈ X,
there exists an open nbd W of x such that U γ ∩ V γ ⊇ W γ .
Definition [14]. An operation γ on τ is said to be open, if for every nbd U of each x ∈ X ,
there exists γ-open set B such that x ∈ B and U γ ⊇ B.
We defined [9] γ∗ -semi-open sets using γ-open sets in the sense of H. Ogata [14] as :
Definition [9]. A subset A of a space (X,τ ) is said to be a γ∗ -semi-open set, if there exists a γ
-open set O such that O ⊆ A ⊆ clγ (O). The set of all γ∗ -semi-open sets is denoted by SOγ∗ (X ).
Definition [5]. A function f : (X , τ) → (Y, τ) is said to be γ-semi-continuous if for any γ-open
set B of Y, f −1 (B) is γ∗ -semi-open in X.
Definition [5]. A function f : X → Y is said to be γ-semi-open (closed) if for each γ-open
(closed) set U in X, f (U) is γ∗ -semi-open (closed) in Y.
Definition [5]. A set A in a space X is said to be γ∗ -semi-closed if there exists a γ-closed set F
such that int γ (F ) ⊆ A ⊆ F .
Proposition [5]. A subset A of a space X is γ∗ -semi-closed if X − A is γ∗ -semi-open.
Definition [10]. A subset A of a space X is said to be γ-semi-nbd of a point x ∈ X if there
exists a γ∗ -semi-open set U such that x ∈ U ⊆ A.
3. γ∗ -irresolute functions
Definition 3.1. Let X and Y be spaces. A function f : X → Y is said to be γ∗ -irresolute if
and only if for any γ∗ -semi-open subset S of Y, f −1 (S) is γ∗ -semi-open in X.
B. Ahmad, S. Hussain , and T. Noiri / Eur. J. Pure Appl. Math, 1 (2008), (22-29)
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Definition 3.2[5]. An operation γ is said to be semi-regular, if for any semi-open sets U and
V containing x ∈ X , there exists a semi-open set W containing x such that U γ ∩ V γ ⊇ W γ .
Theorem 3.3. A function f : X → Y is γ∗ -irresolute if and only if for each x in X, the inverse
of every γ-semi-nbd of f (x) is a γ-semi-nbd of x, where γ is a semi regular operation .
Proof. Let x ∈ X and A a γ-semi-nbd of f (x) . By definition of γ-semi-nbd , there exists
V ∈ SOγ∗ (Y ) such that f (x) ∈ V ⊆ A. This implies that x ∈ f −1 (V ) ⊆ f −1 (A). Since f
is γ∗ -irresolute, so f −1 (V ) ∈ SOγ∗ (X ). Hence f −1 (A) is a γ-semi-nbd of x. This proves the
necessity.
Conversely, let A ∈ SOγ∗ (Y ) . Put B = f −1 (A) . Let x ∈ B , then f (x) ∈ A . Clearly,
A ( being γ∗ -semi-open) is a γ-semi-nbd of f (x) . So by hypothesis, B = f −1 (A) is a γ-seminbd of
S x. Hence by definition , there exists B x ∈ SOγ∗ (X ) such that ∗ x ∈ B x ⊆ B . Thus
B = x∈B B x . Since γ is semi regular, therefore it follows that B is γ -semi-open in X [5].
Therefore f is γ∗ -irresolute.
Remark 3.4. γ-semi-nbd of x may be replaced by γ∗ -semi-open nbd of x in Theorem 3.3.
Theorem 3.5. A function f : X → Y is γ∗ -irresolute if and only if for each x in X and each
γ-semi-nbd A of f (x) , there is a γ-semi-nbd B of x such that f (B) ⊆ A , where γ is a semi
regular operation.
Proof. Let x ∈ X and A a γ-semi-nbd of f (x) . Then there exists O f (x) ∈ SOγ∗ (Y ) such that
f (x) ∈ O f (x) ⊆ A . It follows that x ∈ f −1 (O f (x) ) ⊆ f −1 (A). By hypothesis,
f −1 (O f (x) ) ∈ SOγ∗ (X ) . Let B = f −1 (A). Then it follows that B is γ-semi-nbd of x and
f (B) = f f −1 (A) ⊆ A . This proves the necessity.
Conversely, let U ∈ SOγ∗ (Y ) . Take O = f −1 (U) . Let x ∈ O , then f (x) ∈ U . Thus U is a
γ-semi-nbd of f (x) . So by hypothesis, there exists a γ-semi-nbd Vx of x such that
f (Vx ) ⊆ U . Thus it follows that x ∈ Vx ⊆ f −1 f (Vx ) ⊆ f −1 (U) = O . So Vx is a γ-semi-nbd
S
of x, which implies there exists an Ox ∈ SOγ∗ (X ) such that x ∈ Ox ⊆ O. Thus O = x∈o Ox .
Since γ is semi regular, then it follows that O is γ∗ -semi-open in X. Thus f is γ∗ -irresolute.
Definition 3.6[10]. Let X be a space. A ⊆ X and p ∈ X . Then p is called a γ∗ -semi-limit point
of A if U ∩ (A− {p}) 6= φ , for any γ∗ -semi-open set U containing p. The set of all γ∗ -semi-limit
points of A is called a γ∗ -semi-derived set of A and is denoted by sdγ∗ (A) .
Clearly if A ⊆ B then sdγ∗ (A) ⊆ sdγ∗ (B) ..... (*)
Denote Γ(X ), the set of all monotone operators on X. Then we have:
Definition 3.7[9]. Let A be a subset of space X and γ ∈ Γ(X ). The intersection of all γ∗ -semiclosed sets containing A is called γ∗ -semi-closure of A and is denoted by sclγ∗ (A) .
Remark 3.8. From the definition 3.6, it follows that p is a γ∗ -semi-limit point of A if and only
if p ∈ sclγ∗ (A − {p}) .
Theorem 3.9[10]. For any A, B ⊆ X , the γ∗ -semi-derived sets have the following properties:
(1) scl
S γ∗ (A) = A ∪ sdγ∗ (A).
S
(2) i sdγ∗ (Ai ) = sdγ∗ ( i Ai ) .
(3) sdγ∗ (sdγ∗ (A)) ⊆ sdγ∗ (A).
(4) sclγ∗ (sdγ∗ (A)) = sdγ∗ (A) .
In terms of γ∗ -semi-derived sets, we use Theorem 3.9 and characterize γ∗ -irresolute func-
B. Ahmad, S. Hussain , and T. Noiri / Eur. J. Pure Appl. Math, 1 (2008), (22-29)
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tions as:
Theorem 3.10. A function f : X → Y is γ∗ -irresolute if and only if
f (sdγ∗ (A)) ⊆ sclγ∗ ( f (A)), for all A ⊆ X .
Proof. Let f : X → Y be γ∗ -irresolute. Let A ⊆ X , and x ∈ sdγ∗ (A). Assume that f (x) ∈
/ f (A)
∗
and let V denote a γ-semi-nbd of f (x). Since f is γ -irresolute, so by Theorem 3.5, there exists
a γ-semi-nbd U of x such that f (U) ⊆ V . From x ∈ sdγ∗ (A), it follows that U ∩ A 6= φ ; there
exists, therefore, at least one element a ∈ U ∩ A such that f (a) ∈ f (A) and f (a) ∈ V . Since
f (x) ∈
/ f (A), we have f (a) 6= f (x). Thus every γ-semi-nbd of f (x) contains an element f (a)
of f (A) different from f (x). Consequently, f (x) ∈ sdγ∗ ( f (A)).This proves the necessity.
Conversely, suppose that f is not γ∗ -irresolute. Then by Theorem 3.5, there exists x ∈ X
and a γ-semi-nbd V of f (x) such that every γ-semi-nbd U of x contains at least one element
a ∈ U for which f (a) ∈
/ V . Put A = {a ∈ X : f (a) ∈
/ V } . Since f (x) ∈ V , therefore x ∈
/ A and
hence f (x) ∈
/ f (A) . Since f (A) ∩ (V − f (x)) = φ , therefore f (x) ∈
/ sdγ∗ ( f (A)). It follows that
f (x) ∈ f (sdγ∗ (A)) − ( f (A) ∪ sdγ∗ ( f (A))) 6= φ , which is a contradiction to the given condition.
This proves sufficiency.
Theorem 3.11. Let f : X → Y be one-to-one function. Then f is γ∗ -irresolute if and only if
f (sdγ∗ (A)) ⊆ sdγ∗ ( f (A)), for all A ⊆ X .
Proof. Let A ⊆ X , x ∈ sdγ∗ (A) and V be a γ-semi-nbd of f (x). Since f is γ∗ -irresolute, so by
Theorem 3.5, there exists a γ-semi-nbd U of x such that f (U) ⊆ V . But x ∈ sdγ∗ (A) gives there
exists an element a ∈ U ∩ A such that a 6= x. Clearly f (a) ∈ f (A) and since f is one to one,
f (a) 6= f (x) . Thus every γ-semi-nbd V of f (x) contains an element f (a) of f (A) different
from f (x). Consequently f (x) ∈ sdγ∗ ( f (A)). Therefore we have f (sdγ∗ (A)) ⊆ sdγ∗ ( f (A)) .
This proves the necessity.
Sufficiency follows from Theorem 3.10. This completes the proof.
H. Ogata [14] defined the notion of γ-T2 spaces in topological spaces, we generalize this
notion and define:
Definition 3.12. A space X is said to be γ-semi-T2 , if for each two distinct points x, y ∈ X
there exist U, V ∈ SOγ∗ (X ) such that x ∈ U, y ∈ V and U ∩ V = φ .
Theorem 3.13. If f : X → Y is a γ∗ -irresolute injection and Y is γ-semi-T2 , then X is
γ-semi-T2 .
Proof. Let x 1 and x 2 be two distinct points of X. Since f is injective, therefore
f (x 1 ) 6= f (x 2 ) . Since Y is γ-semi-T2 , there exist V1 , V2 ∈ SOγ∗ (Y ) such that f (x 1 ) ∈ V1 ,
f (x 2 ) ∈ V2 and V1 ∩ V2 = φ. Then x 1 ∈ f −1 (V1 ), x 2 ∈ f −1 (V2 ) and f −1 (V1 ) ∩ f −1 (V2 ) = φ.
Since f is γ∗ -irresolute, so f −1 (V1 ), f −1 (V2 ) ∈ SOγ∗ (X ). This proves that X is γ-semi-T2 .
4. γ-pre-semi-open functions
Definition 4.1. Let X and Y be spaces. Then a function f : X → Y is said to be γ-pre-semiopen if and only if for each A ∈ SOγ∗ (X ) , f (A) ∈ SOγ∗ (Y ) .
B. Ahmad, S. Hussain , and T. Noiri / Eur. J. Pure Appl. Math, 1 (2008), (22-29)
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Remark 4.2. The class of γ-pre-semi-open functions is a subclass of class of γ-semi-open
functions defined in [5].
Note that if f : X → Y and g : Y → Z be any two γ-pre-semi-open functions, then the
composition g o f : X → Z is a γ-pre-semi-open function.
The following theorem is easy to prove:
Theorem 4.3. A function f : X → Y is γ-pre-semi-open if and only if for each x ∈ X and for
every A ∈ SOγ∗ (X ) such that x ∈ A, there exists B ∈ SOγ∗ (Y ) such that f (x) ∈ B and B ⊆ f (A).
Theorem 4.4. A function f : X → Y is γ-pre-semi-open if and only if for each x ∈ X and for
every γ-semi-nbd U of x in X, there exists a γ-semi-nbd V of f (x) in Y such that V ⊆ f (U) .
Proof. Let U be a γ-semi-nbd of x ∈ X . Then by definition there exists W ∈ SOγ∗ (X ) such
that x ∈ W ⊆ U . Then f (x) ∈ f (W ) ⊆ f (U) . Since f is γ-pre-semi-open , therefore
f (W ) ∈ SOγ∗ (Y ) . Hence V = f (W ) is a γ-semi-nbd of f (x) and V ⊆ f (U) .
Conversely, let U ∈ SOγ∗ (X ) and x ∈ U . Then U is a γ-semi-nbd of x. So by hypothesis,
there exists a γ-semi-nbd V of f (x) such that f (x) ∈ V ⊆ f (U). That is, f (U) is a γ-semi-nbd
of f (x) . Thus f (U) is a γ-semi-nbd of each of its points. Thus f (U) is γ∗ -semi-open. Hence f
is γ-pre-semi-open. This completes the proof.
Definition 4.5[9]. Let A be a subset of a space X and γ ∈ Γ(X ). The union of all γ∗ -semi-open
sets of X contained in A is called γ∗ -semi-interior of A and is denoted by sint γ∗ (A) .
Theorem 4.6. A function f : X → Y is γ-pre-semi-open if and only if
f (sint γ∗ (A)) ⊆ sint γ∗ ( f (A)), for all A ⊆ X .
Proof. Let x ∈ sint γ∗ (A). Then there exists U ∈ SOγ∗ (X ) such that x ∈ U ⊆ A. So f (x) ∈
f (U) ⊆ f (A). Since f is γ-pre-semi-open , therefore f (U) is γ∗ -semi-open in Y. Hence f (x) ∈
sint γ∗ ( f (A)) . Thus f (sint γ∗ (A)) ⊆ sint γ∗ ( f (A)) . This proves necessity.
Conversely let U ∈ SOγ∗ (X ) . Then by hypothesis, f (U) = f (sint γ∗ (U)) ⊆ sint γ∗ ( f (U)) ⊆
f (U) or f (U) ⊆ sint γ∗ ( f (U)) ⊆ f (U) . This implies f (U) is γ∗ -semi-open in Y. So f is γ-presemi-open.
We use Theorem 4.6 and prove:
Theorem 4.7. A function f : X → Y is γ-pre-semi-open if and only if
sint γ∗ ( f −1 (A)) ⊆ f −1 (sint γ∗ (A)), for all A ⊆ X .
Proof. Let A be any subset of Y. Clearly, sint γ∗ ( f −1 (A)) is γ∗ -semi-open in Y. Also f (sint γ∗ ( f −1 (A)) ⊆
f ( f −1 (A)) ⊆ A . Since f is γ-pre-semi-open, by Theorem 4.6 we have
f (sint γ∗ ( f −1 (A)) ⊆ sint γ∗ (A) . Therefore,
sint γ∗ ( f −1 (A)) ⊆ f ( f −1 (sint γ∗ ( f −1 (A)))) ⊆ f −1 (sint γ∗ (A)) or
sint γ∗ ( f −1 (A)) ⊆ f −1 (sint γ∗ (A)). This proves necessity.
Conversely, let B ⊆ X . By hypothesis, we obtain
sint γ∗ (B) ⊆ sint γ∗ ( f −1 f (B)) ⊆ f −1 (sint γ∗ f (B)) . This implies that f (sint γ∗ (B)) ⊆ f (sint γ∗ ( f −1 f (B))) ⊆
f ( f −1 (sint γ∗ f (B))) ⊆ sint γ∗ ( f (B)). Consequently, f (sint γ∗ (B)) ⊆ sint γ∗ ( f (B)) , for all
B ⊆ X . By Theorem 4.6 , f is γ-pre-semi-open.
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We use Theorem 4.7 and prove:
Theorem 4.8. A function f : X → Y is γ-pre-semi-open if and only if
f −1 (sclγ∗ (A)) ⊆ sclγ∗ ( f −1 (A)), for all A ⊆ X .
Proof. Let A ⊆ Y . By Theorem 4.7 , sint γ∗ ( f −1 (Y − A)) ⊆ f −1 (sint γ∗ (Y − A)) . This implies
that sint γ∗ (X − f −1 (A)) ⊆ f −1 (sint γ∗ (Y − A)) . As sint γ∗ (A) = X − sclγ∗ (X − A) [5], therefore
X − sclγ∗ ( f −1 (A)) ⊆ f −1 (Y − sclγ∗ (A)). or X − sclγ∗ ( f −1 (A)) ⊆ X − f −1 (sclγ∗ (A)). Hence
f −1 (sclγ∗ (A)) ⊆ sclγ∗ ( f −1 (A)). This proves necessity.
Conversely, let A ⊆ Y . By hypothesis, f −1 (sclγ∗ (Y − A)) ⊆ sclγ∗ ( f −1 (Y − A)) . This
implies X − sclγ∗ ( f −1 (Y − A)) ⊆ X − f −1 (sclγ∗ (Y − A)) . Hence X − sclγ∗ (X − f −1 (A)) ⊆
f −1 (Y − (sclγ∗ (Y − A))). This gives sint γ∗ ( f −1 (A)) ⊆ f −1 (sint γ∗ (A)) . Now from Theorem 4.7,
it follows that f is γ-pre-semi-open. This completes the proof.
Definition 4.9[14]. A function f : (X , τ) → (Y, τ) is said to be (γ, β) -continuous, if for each
x ∈ X and each open set V containing f (x) , there exists an open set U such that x ∈ U and
f (U γ ) ⊆ V β , where γ and β are operations on τ and δ respectively.
Definition 4.10[14]. A function f : (X , τ) → (Y, τ) is said to be (γ, β)-open (closed), if for
any γ-open (closed) set A of X, f (A) is β-open(closed) in Y.
In [10], we proved the following Theorem:
Theorem 4.11[10]. If f : X → Y is a (γ, β)-open and (γ, β)-continuous function, then
f −1 (B) ∈ SOγ∗ (X ) , for every B ∈ SOβ ∗ (Y ) , where β is an open operation.
We use this theorem and prove:
Theorem 4.12. Let X, Y and Z be three spaces, and let f : X → Y and g : Y → Z be two
functions such that g o f : X → Z a γ-pre-semi-open function. Then
(1) If f is a (γ, β)-open and (γ, β)-continuous surjection, then g is β -pre-semi-open.
(2) If g is a (β, δ)-open and (β, δ)-continuous injection, then f is γ-pre-semi-open.
Proof.(1). Let V be an arbitrary γ∗ -semi-open set in Y. Since f is a (γ, β)-open and (γ, β)continuous, then by Theorem 4.11, f −1 (V ) is a γ∗ -semi-open set in X. Also g o f is γ-pre-semiopen and f is surjection, we have g(V ) = (g o f )( f −1 (V )) is a β ∗ -semi-open set in Z. This
proves that g is β-pre-semi-open.
(2). Since g is injective so for every subset A of X, f (A) = g −1 (g( f (A))). Let U be an arbitrary
γ∗ -semi-open set in X. Then (g o f )(U) is γ∗ -semi-open. Since g is a (β, δ)-open and (β, δ)semi-continuous, therefore by above Theorem 4.11, f (U) is γ∗ -semi-open in Y. This shows
that f is γ-pre-semi-open.
Theorem 4.13. Let f : X → Y and g : Y → Z be two functions such that g o f : X → Z
is γ∗ -irresolute. Then
(1) If g is a β-pre-semi-open injection, then f is γ∗ -irresolute.
(2) If f is a γ-pre-semi-open surjection, then g is β ∗ -irresolute.
Proof.(1). Let U ∈ SOβ ∗ (Y ) . Since g is β-pre-semi-open, then g(U) ∈ SOδ∗ (Z). Also g o f is
γ∗ -irresolute, and therefore; (g o f )−1 (g(U)) ∈ SOγ∗ (X ) . Since g is injective (g o f )−1 (g(U)) =
( f −1 o g −1 )g(U) = f −1 (g −1 g(U)) = f −1 (U). Consequently, f −1 (U) is γ∗ -semi-open in X. This
B. Ahmad, S. Hussain , and T. Noiri / Eur. J. Pure Appl. Math, 1 (2008), (22-29)
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proves that f is γ∗ -irresolute. This proves (1).
(2). Let V ∈ SOδ∗ (Z) . Since g o f is γ∗ -irresolute, then (g o f )−1 (V ) ∈ SOγ∗ (X ). Also f
is γ-pre-semi-open, so f (g o f )−1 (V ) is β ∗ -semi-open in Y. Since f is surjective, we obtain
f o(g o f )−1 (V ) = f o( f −1 o g −1 )(V ) = ( f o f −1 )o g −1 (V ) = g −1 (V ). It follows that g −1 (V ) ∈
SOγ∗ (Y ) . This proves that g is a β ∗ -irresolute function.
5. γ-pre-semi-closed functions
Definition 5.1. A function f : X → Y is γ-pre-semi-closed if and only if the image set f (A)
is γ∗ -semi-closed, for each γ∗ -semi-closed subset A of X.
It is obvious that the composition of two γ-pre-semi-closed mappings is a γ-pre-semi-closed
mapping.
Theorem 5.2. A function f : X → Y is γ-pre-semi-closed if and only if
sclγ∗ f (B) ⊆ f (sclγ∗ (B)), for every subset B of X.
Proof. Suppose f is γ-pre-semi-closed and let B ⊆ X . Since f is γ-pre-semi-closed , therefore
f (sclγ∗ (B)) is γ∗ -semi-closed in Y. Since f (B) ⊆ f (sclγ∗ (B)) , we obtain
sclγ∗ ( f (B)) ⊆ f (sclγ∗ (B)) . This proves necessity.
Conversely, suppose A is a γ∗ -semi-closed set in X. By hypothesis, we obtain f (A) ⊆
sclγ∗ ( f (A)) ⊆ f (sclγ∗ (A)) = f (A). Hence f (A) = sclγ∗ ( f (A)). Thus f (A) is γ∗ -semi-closed
set in Y. This proves that f is γ-pre-semi-closed.
Theorem 5.3. A function f : X → Y is γ-pre-semi-closed if and only if
int γ (clγ ( f (B))) ⊆ f (sclγ∗ (B)), for every subset B of X.
Proof. Suppose f is γ-pre-semi-closed and B is any subset of X. Then f (sclγ∗ (B)) is γ∗ semi-closed in Y. This implies that there exists a γ-closed subset A of Y such that int γ (A) ⊆
f (sclγ∗ (B)) ⊆ A . This gives that int γ clγ ( f (sclγ∗ (B))) ⊆ int γ (A) ⊆ f (sclγ∗ (B)). Then int γ (clγ ( f (B)) ⊆
int γ (clγ ( f (sclγ∗ (B)))) gives int γ (clγ ( f (B))) ⊆ f (sclγ∗ (B)) . This proves necessity.
Conversely, suppose A is a γ∗ -semi-closed set in X. Then by hypothesis, int γ (clγ ( f (A)) ⊆
f (sclγ∗ (A)) = f (A) . Put B = clγ ( f (A)) . Clearly, B is γ-closed in Y. This implies that int γ (B) ⊆
f (A) ⊆ B. Hence f (A) is γ∗ -semi-closed in Y. This implies f is γ-pre-semi-closed.
Theorem 5.4. A bijective function f : X → Y is a γ-pre-semi-closed if and only if for each
subset A of Y and each γ∗ -semi-open set B in X containing f −1 (A) , there exists a γ∗ -semi-open
set C in Y containing A such that f −1 (C) ⊆ B , where γ is a semi regular operation.
Proof. Let C = Y − f (X − B) . Then C c = f (B c ) . Since f is γ-pre-semi-closed, so C is
γ∗ -semi-open. Since f −1 (A) ⊆ B, we have C c = f (B c ) ⊆ f ( f −1 (Ac )) ⊆ Ac . Hence, A ⊆ C,
and thus C is a γ-semi-open nbd of A. Further B c ⊆ f −1 ( f (B c )) = f −1 (C c ) = ( f −1 (C))c . This
proves that f −1 (C) ⊆ B .
Conversely, suppose F is a γ∗ -semi-closed set in X. Let y ∈ Y − f (F ) . Then f −1 ( y) ⊆
X − f −1 ( f (F )) ⊆ X − F and X − F is γ∗ -semi-open in X. Hence by hypothesis, there exists
a γ∗ -semi-open set C containing y such that f −1 (C) ⊆ X − F . Since f is one-one, we have
REFERENCES
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S
y ∈ C ⊆ Y − f (F ) . Thus Y − f (F ) = y∈Y − f (F ) C . Hence Y − f (F ) is γ∗ -semi-open set [5].
This proves that f is γ-pre-semi-closed.
We use Theorem 4.8 and give the following characterizations:
Theorem 5.5. Let f : X → Y be a bijective function. Then the following are equivalent:
(1) f is γ-pre-semi-closed.
(2) f is γ-pre-semi-open.
(3) f −1 is γ∗ -irresolute.
Proof. (1) ⇒ (2) It is straightforward.
(2) ⇒ (3). Let A ⊆ X . Since f is γ-pre-semi-open, by Theorem 4.9, f −1 (sclγ∗ ( f (A))) ⊆
sclγ∗ ( f −1 f (A)) implies sclγ∗ ( f (A)) ⊆ f (sclγ∗ (A)). Thus sclγ∗ ( f −1 )−1 (A) is contained in
( f −1 )−1 (sclγ∗ (A)) , for every subset A of X. Then again by Theorem 4.8 , it follows that f −1 is
γ∗ -irresolute.
(3) ⇒ (1). Let A be a γ∗ -semi-closed set in X. Then X − A is γ∗ -semi-open in X. Since f −1 is
γ∗ -irresolute, ( f −1 )−1 (X − A) is γ∗ -semi-open in Y. But ( f −1 )−1 (X − A) = f (X − A) = Y − f (A).
Thus f (A) is γ∗ -semi-closed in Y. This proves that f is γ-pre-semi-closed. This completes the
proof.
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